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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Semilattices of finitely generated ideals of exchange rings with finite stable rank


Author: F. Wehrung
Journal: Trans. Amer. Math. Soc. 356 (2004), 1957-1970
MSC (2000): Primary 06A12, 20M14, 06B10; Secondary 19K14
DOI: https://doi.org/10.1090/S0002-9947-03-03369-5
Published electronically: October 28, 2003
MathSciNet review: 2031048
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Abstract:

We find a distributive $(\vee ,0,1)$-semilattice $S_{\omega _1}$ of size $\aleph _1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular:

  • [—] There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to $S_{\omega _1}$.

  • [—] There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $S_{\omega _1}$.

  • These results are established by constructing an infinitary statement, denoted here by $\mathrm {URP_{sr}}$, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice $S_{\omega _1}$.


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    Additional Information

    F. Wehrung
    Affiliation: Département de Mathématiques, CNRS, UMR 6139, Université de Caen, Campus II, B.P. 5186, 14032 Caen Cedex, France
    MR Author ID: 242737
    Email: wehrung@math.unicaen.fr

    Keywords: Semilattice, distributive, monoid, refinement, ideal, stable rank, strongly separative, exchange ring, lattice, congruence
    Received by editor(s): January 3, 2003
    Received by editor(s) in revised form: April 2, 2003
    Published electronically: October 28, 2003
    Article copyright: © Copyright 2003 American Mathematical Society